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5 EQUICONTINUOUS SETS 141

2. Let n -> r_n be a bijection of N onto the set of rational numbers in the interval [0,1 ] = I.
Define by induction a sequence (I_n) of closed intervals contained in I, such that: (1) the
center of !„ is r_kn, where k_n is the smallest index p such that r_p is not in the union of
the intervals I with h<n\(2) the length of !„ is < 1/4", and I_n does not meet any of the
I_ft with h < n. In the product space I x R, define a bounded real continuous function u
having the following properties: (1) for each integer n ^ 0, jc -> u(x, n) takes the value 1
for x = r_fcn, is equal to 0 for x $ I_n, and 0 ^ u(x, n) ^ 1 for all jc e I; (2) for each x e I,
the function y^u(x₉y) has the form ay + J3 in each of the intervals ]-oo, 0[ and
[«, n + 1] (n e N). Show that there is no finite system of functions v_t e ^_R(l), w_t ^e ^a(R)

(1 < i ^ n) such that u(x,y) — £ ^OMOO < 1/4 in I X R. (Suppose the contrary;

t = i

consider the functions u_n: x -> w(jc, «) in ^_R(I), and observe that \\u_a \\ = 1, \\u_n ~ u_m \\ = 1
for m T& n. If there existed a finite dimensional subspace E of ^_R(I) such that d(u_n, E) ^ 1/4
for each n, there would exist in E an infinite sequence (/*„) such that PJ| = 2 and
\\h_n — h_m\\'& 1/2 for m^n, contradicting (5.10.1).)

3. Let E be the interval [0,1] in R.

(a) Show that if a_k (1 ^ k ^ n) are n distinct points of E, the functions x-*\x — a_k\
are linearly independent in ^_R(E).

(b) Deduce from (a) that the function (x, y) -* x — y\ in E X E cannot be written as

^ finite sum ]T Vi(x)wi(y), where Vi and w_t are continuous in E.

4. Show that the Banach space ^jf (R) is not separable. (Use a similar method as the one
applied in the Problem of Section 5.10.)

5. EQUICONTINUOUS SETS

Let H be a subset of the space J^t_F(E) (E metric space, F normed space);
we say that H is equicontinuous at a point X_Q e E if, for any s > 0, there is a
8 > 0 such that the relation d(x₀ ,x)^8 implies \\f(x) —f(x_Q)\\ < sfor every
/e H (the important thing here being that d is independent off). We say that
H is equicontinuous if it is equicontinuous at every point of E.

Examples

(7.5.1) Suppose there exist two constants c, a > 0 such that ||/(x) -f(y)\\ <S
c - (d(x, y))* for anyfe H, and any pair of points x, y of E; then H is equi-
continuous.

(7.5.2) Any finite set of functions which are continuous at a point x₀ (resp.
in E) is equicontinuous at x₀ (resp. equicontinuous). More generally any
finite union of sets of functions which are equicontinuous at x₀ (resp. equi-
continuous) is equicontinuous at :x;₀ (resp. equicontinuous).



	5 EQUICONTINUOUS SETS 141 2. Let n -> r_n be a bijection of N onto the set of rational numbers in the interval [0,1 ] = I. Define by induction a sequence (I_n) of closed intervals contained in I, such that: (1) the center of !„ is r_kn, where k_n is the smallest index p such that r_p is not in the union of the intervals I with h<n\(2) the length of !„ is < 1/4", and I_n does not meet any of the I_ft with h < n. In the product space I x R, define a bounded real continuous function u having the following properties: (1) for each integer n ^ 0, jc -> u(x, n) takes the value 1 for x = r_fcn, is equal to 0 for x $ I_n, and 0 ^ u(x, n) ^ 1 for all jc e I; (2) for each x e I, the function y^u(x₉y) has the form ay + J3 in each of the intervals ]-oo, 0[ and [«, n + 1] (n e N). Show that there is no finite system of functions v_t e ^_R(l), w_t ^e ^a(R) n (1 < i ^ n) such that u(x,y) — £ ^OMOO < 1/4 in I X R. (Suppose the contrary; t = i consider the functions u_n: x -> w(jc, «) in ^_R(I), and observe that \\u_a \\ = 1, \\u_n ~ u_m \\ = 1 for m T& n. If there existed a finite dimensional subspace E of ^_R(I) such that d(u_n, E) ^ 1/4 for each n, there would exist in E an infinite sequence (/„) such that PJ\| = 2 and \\h_n — h_m\\'&* 1/2 for m^n, contradicting (5.10.1).) 3. Let E be the interval [0,1] in R. (a) Show that if a_k (1 ^ k ^ n) are n distinct points of E, the functions x-\x — a_k\* are linearly independent in ^_R(E). (b) Deduce from (a) that the function (x, y) - x — y\* in E X E cannot be written as n ^ finite sum ]T Vi(x)wi(y), where Vi and w_t are continuous in E. 4. Show that the Banach space ^jf (R) is not separable. (Use a similar method as the one applied in the Problem of Section 5.10.)

	5. EQUICONTINUOUS SETS Let H be a subset of the space J^t_F(E) (E metric space, F normed space); we say that H is equicontinuous at a point X_Q e E if, for any s > 0, there is a 8 > 0 such that the relation d(x₀ ,x)^8 implies \\f(x) —f(x_Q)\\ < sfor every /e H (the important thing here being that d is independent off). We say that H is equicontinuous if it is equicontinuous at every point of E.

	Examples (7.5.1) Suppose there exist two constants c, a > 0 such that \|\|/(x) -f(y)\\ <S c - (d(x, y))* for anyfe H, and any pair of points x, y of E; then H is equi- continuous. (7.5.2) Any finite set of functions which are continuous at a point x₀ (resp. in E) is equicontinuous at x₀ (resp. equicontinuous). More generally any finite union of sets of functions which are equicontinuous at x₀ (resp. equi- continuous) is equicontinuous at :x;₀ (resp. equicontinuous).